Understanding the success of Page's model and related empirical equations in fitting experimental data of diffusion phenomena in food matrices

Abstract

Page's equation is one of the most successful empirical equations applied to describe water migration throughout food drying processes. However, when analyzed in detail, the mathematical derivation shows a number of inconsistencies.The hypothesis is that the noteworthy success of Page's equation could be explained by its mathematical similarity with the solution of an anomalous diffusion phenomenon. In addition, the extraordinary success of Page's equation has induced other researchers to search for similar equations, but without a phenomenological comprehension.The objectives are: 1) demonstrate that Page's equation presents some inconsistences in its mathematical derivation; 2) demonstrate that the goodness of fit of Page's equation could be explained by its similarity with the mathematical solution of anomalous diffusion model.A comprehensive examination of Page's thesis was performed by analyzing its theoretical development and experimental data. Also, data reported in the literature were analyzed to test the application of Page's equation, Fick's second law and anomalous diffusion model.This study demonstrates that Page's equation was raised from a misinterpretation of Fick's second law. In addition, anomalous diffusion solution has remarkable similarities with Page's equation, mainly because of the inclusion of time exponential parameter (n). One important outcome from the apple drying data was that n must be constant and independent of temperature, given that n is related with the microstructure of the food matrix. In conclusion, the mathematical solution of anomalous diffusion model through the fractional calculus approach may, in part, explain the goodness of fit of Page's equation to drying data.